The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: imaginary numbers

The concept of imaginary numbers has always been a fascinating one. The Greek mathematician Heron of Alexandria, born around 10 AD, is noted as being the first person to have come up with the idea of imaginary numbers. It wasn’t until the 1500’s, though, that rules for arithmetic and notation for complex numbers really came to fruition. Of course, at the time most people thought imaginary numbers were just stupid and pointless. Heck, today I’m pretty sure most people still think imaginary and complex numbers are stupid and pointless. Surely they can be used for more than just generating pretty looking fractals (like the Mandelbrot set), right?

Yes, because of imaginary numbers there is a solution to any type of polynomial equation… but there has to be more use to them than that, right? The topic I wish to present in this article is about some of the other applications of imaginary numbers. Imaginary numbers are really useful and they can be used to do all sorts of awesome things!

While presenting this information, I do not claim to list every single practical use of imaginary numbers. There are many useful applications that involve some crazy complicated mathematics and are admittedly beyond the scope of my understanding at the present time. Rather, I wish to share a few of my favorite applications of imaginary numbers. It is my hope that the reader will learn more about why mathematicians have studied so much about imaginary and complex numbers.

First, complex numbers have a remarkable application in triangular geometry. There is a fascinating theorem called “Marden’s theorem”. I read about this theorem in an article written by Dan Kalman, a doctor of mathematics who works in the Department of Mathematics and Statistics at American University. He claims that this theorem is “the Most Marvelous Theorem in Mathematics.”

A visualization of a Steiner inellipse with its foci. The ellipse is based on the polynomial p(z)=z3-(9+9i)z2+(3+52i)z+(33-39i). The black dots are the zeros of p(z), and the red dots are the zeroes of p'(z) and the foci of the inellipse. Uploaded by User Kmhkmh for Wikipedia on 2/6/2010. Creative Commons license. Reuse permitted.

Basically, this theorem can help one find the foci of a Steiner inelipse. A Steiner inellipse is simply an ellipse that is inside of a triangle and is tangent to the midpoints of the three sides of the triangle. Such an ellipse is shown in the following diagram.

The foci of a Steiner inellipse can be found by using complex numbers! The triangle’s vertices can be written as points in the complex plane as follows: a = xA + yAi, b = xB + yBi, and c = xC + yCi. Marden’s theorem states that if you take the derivative of the cubic equation (x-a)(x-b)(x-c) = 0 and set it to zero, then the solutions of this equation will be the two foci of the Steiner inellipse in complex numbers. Isn’t that a really bizarre theorem? If you think about it, though, it makes some intuitive sense. When you take the derivative of an equation and set it equal to zero, the solutions of that equation give you the maximum and minimum values found on the arcs in the equation. A regular cubic equation could have up to two arcs, so it’s natural that there would be two max/min values. The fact that these two values are the two foci of the inellipse is really interesting.

As it turns out, using complex numbers here gives us a very amazing and useful geometric tool to use. There are also a few generalizations of this theorem that apply to different types of polynomials and other geometric shapes!

So it seems that first we have geometric applications for complex numbers. Now I would like to present a second category of applications. These are related to phasor calculus. A phasor is a complex number that represents a sinusoidal function. Thanks to the amazing Euler’s formula (eiπ = cosx + sinx), sinusoidal functions can be rewritten as complex numbers. This allows for easier problem solving and analysis for many types of problems.

For instance, in electrical engineering alternating currents can be a pain to analyze sometimes. After all, they have voltages that exhibit sinusoidal behavior. With the use of phasors, one can analyze aspects of AC circuits more easily. Analysis of resistors, capacitors, and inductors can be combined into a single complex number, which is called the impedance. Phasors are comparatively easy to interpret, so it’s a lot easier to study AC circuits when studying them in the complex plane! In addition to AC circuits, complex numbers are similarly useful when studying electromagnetic fields, where the quantities of electric and magnetic field strength are combined into a single complex number.

The last application I wish to bring up involves the usage of imaginary numbers to solve integration problems. As it turns out, we can use the aforementioned Euler’s formula to simplify real integration problems and help us find real answers. This is done by using a base integral that has a complex solution. An example of a base integral would be∫ e(1+i)xdx. Using simple u substitution, we can find the answer to this integral, which is ((1-i)/2)e(1+i)x + c1 + ic2. With this known imaginary answer, we can compute the answer to a real integral.

Consider, for example,∫ excosxdx. First, we rewrite the previously mentioned base integral as: ∫ exeixdx. Then we can use Euler’s formula to alter this integral further:∫ exeixdx = ∫ ex(cosx + isinx)dx. This will further simplify to∫ excosxdx + i∫ exsinxdx. We can set the known solution of the base integral equal to this complex integral and solve for ∫ excosxdx , which is the real integral we are trying to compute. We will see that the imaginary parts must be equal and the real parts must also be equal. Solving in this manner will show us that ∫ excosxdx = .5ex(cosx + sinx) + c. Hopefully I don’t have to explain how useful integrals are! The fact that complex numbers can help us solve integrals alone means they are really useful.

I think in general it seems that whenever there’s an oscillatory phenomenon of any kind then complex numbers are naturally helpful in describing said phenomenon. Complex numbers have multiple substantial applications in a multitude of scientific problems. In addition to the few I’ve mentioned, complex numbers are also used in: quantum mechanics, control theory, signal processing, vibration studies, cartography, and fluid dynamics. Dang. Since a long time ago complex numbers have been thought of as trivial and inconsequential. Descartes himself (who coined the term “imaginary”) called these types of numbers imaginary because he meant for this to be derogatory. However, as we have learned more about math throughout the ages we have found many a useful application for imaginary numbers.

The aforementioned Mandelbrot set. This is a fractal involving a set of complex numbers. Uploaded by User Localhost00 on 10/13/2013. Creative Commons license. Reuse permitted.

It’s kind of embarrassing to admit this as a 24-year-old college student, but I have an imaginary friend. His name is √(-1). He doesn’t keep it as real as some of my other friends, but he’s always my main dude in some of the more complex friendships I get involved in. He’s got a pretty interesting life story, and he’s exceptionally old and wise (imaginary friends never die). He was birthed by the Greek mathematician Heron of Alexandria in the first century A.D. He had a rough childhood (neglected and completely ignored by his father), partially because no one understood him. People ignored him, calling him useless. His name was even meant to be derogatory at one point.4 √(-1) met Gerolamo Cardano (coined the term “imaginary”), his first true friend, in 1637. After Cardano passed away, √(-1) was lonely until he was accepted into a new circle of friends during the 1700 and 1800s: Leonhard Euler, Carl Gauss, and Caspar Wessel. Although his new friends were a little on the nerdy side, they helped restore his personal image – eventually helping him to become a internationally known, and well respected celebrity.

Image: Matheepan Panchalingam, via Flickr.

Okay, enough with my weird metaphorical stories. I’m here to tell you all about imaginary numbers. Now although the previous paragraph was strange and pretty corny, a lot of truth resides within. Heron of Alexandria was the first known person to have encountered imaginary numbers, back in the first century A.D.¹ He has been called a Greek mathematician, but interestingly enough, we now believe he may have actually been Egyptian.¹ So, how did Heron, the father of imaginary numbers, make such an important discovery? Well, he was trying to find the volume of a frustum of a pyramid with a square base. Specifically a pyramid with a side of the lower base is 28, of the upper 4, and the edge 15. Now the formula to solve this problem, as noted in the Stereometria of Heron of Alexandria is:

Using a = 28, b = 4, and c = 15, Heron ultimately reduced this problem to equal √(81 – 144). However instead of writing h = √(-63), he wrote h = √(63).¹ Now whether Heron did this intentionally or not, we’ll never know. What we do know is this: No other scholar even bothered to take notice of these imaginary numbers for at least a thousand years. Believe it or not, from here it took another five hundred years for scholars to start taking imaginary numbers seriously.¹

The Italian engineer and architect Rafael Bombelli started searching for solutions to cubic equations in 1572, and during that time, he started defining some guidelines for using imaginary numbers.5 At this point, most mathematicians didn’t want to believe these numbers were significant, or that they even existed. Bombelli’s “wild idea” – that you could multiply imaginary numbers and end up with a real solution, didn’t sit well with fellow scholars. In fact, Bombelli didn’t even think imaginary numbers could ever have a true purpose, he just viewed them as a useful artifact to solve his equations. Later on, along came Leonhard Euler, a math rockstar in many people’s eyes. Euler really takes the cake here for noticing one of math’s more bizarre imaginary (in a math sense) equations. He showed the world how closely i, e, and π are related and further convinced the math community that i was relevant:2 Today, this is known as Euler’s identity.

eiπ = cosπ +i(sinπ)

Which reduces to:

eiπ = -1 + 0i

In simpler terms:

eiπ = -1

In 1842 Carl Gauss did the impossible. He actually defined imaginary numbers (although Cardano coined the term “imaginary”.).Later on, once the reality of i was more widely accepted, our understanding mathematics expanded greatly. Euler, Gauss, and Wessel (1700s and 1800s) utilized imaginary numbers and led to the creation of complex numbers, which have a vital importance in math/science today.

Complex Numbers:

Instead of describing numbers and algebraic equations as points on a line, complex numbers and equations become points on a plane. Numbers are two dimensional – just like the integer “1″ is the unit distance on the axis of the “real” numbers, “i ” is the unit distance on the axis of the “imaginary” numbers. This results (in general) in complex numbers: They consist of two components, defining their position relative to those two axes. We generally write them as “a + bi” where “a” is the real component, and “b” is the imaginary component.4

A basic graph showing real and imaginary points making up a complex line. Image: Public Domain.

Let me simplify this for you – What this means is that every polynomial equation has roots. Specifically, a polynomial equation in “x” with maximum exponent “n” will always have exactly “n” complex roots.

Me, myself, and i :

Imaginary numbers are more prevalent in the world than you’d think. Without the electronic circuit theories conceived from imaginary numbers, I wouldn’t be typing this blog post, and more importantly, you wouldn’t be reading it. When we plug something into an electrical outlet, we just expect our electronic devices to receive power. In fact, I bet you plugged in your laptop charger today and didn’t even think twice about it. What does exactly does this AC power have to do with imaginary numbers? Everything.

Voltage can be viewed as the amount of force pushing the current through your laptop charger and ultimately into your laptop’s battery. Voltage is a complex number. In fact, if you have a voltage of 110 volts AC at 60 hz (US Standard), that means is that the voltage is a number of magnitude 110. If you were to plot the “real” voltage on a graph with time on the X axis and voltage of the Y, you’d see a basic sine wave. Take the graph below for example:

Basic sine wave showing only real voltage. Image: Public Domain.

If you decided to put your key in the outlet when the voltage was supposedly zero on the real portion of that graph, you’d still get electrocuted. That’s right, even though real voltage is zero here, “imaginary” electricity can give you quite a shock.3 Take the moment marked t1 on the above graph for example. The voltage at time t1 on the complex plane is a point at 110 on the real axis. At time t2, the voltage on the “real” axis is zero – but on the imaginary axis it’s 110. In fact, the magnitude of the voltage is always constant at 110 volts, and this is an important fact to know.5 All our modern day electronics account for these weird voltage patterns, and strangely enough this imaginary unit is represented with a “j ” rather than an i (“i ” is used to represent current)!3

All in all, you can thank Bombelli and Euler for introducing imaginary numbers. Not only do they make math more confusing, but they also have many crucial applications in the world around us today.

Imaginary numbers, which are also known as complex numbers, have had a pretty bad reputation. When most people think of imaginary numbers, they probably break out in a cold sweat from the horrific memories of high school math class. They think that imaginary numbers are utterly incomprehensible and useless in the “real” world. “Imaginary numbers” sound very intimidating to people who are not familiar with them. They also sound highly theoretical with little or no use outside of pure mathematics. In fact, the exact opposite is true.

The most common imaginary number is i, which is formally defined as i = √-1. Since the act of squaring any real number always makes the number positive– whether it began as a negative number or not, it is impossible to find the square root of a negative number without using i. Thus, i made possible an entire class of math problems that were not possible before. For example, √-64 = 8i, cannot be done without using i, because √-64 does not exist in the real number line. Additionally, i can be easily changed from an “imaginary” number into a “real” number simply by squaring it: i² = -1.

The first known person to stumble upon the idea of using an imaginary number to take the square root of a negative number was the Greek mathematician Heron of Alexandria in 50 CE. He was trying to find the volume of a section of a pyramid using a formula that involved the slant height of the pyramid. However, certain values for the slant height would produce the square root of a negative number. Heron was very uncomfortable with this result, so in order to avoid using a negative number, he fudged his calculation by dropping the negative sign.

Girolamo Cardano was an Italian mathematician who was particularly interested in finding the solutions to cubic and quartic equations. In 1545, he published a book titled Ars Magna, which contained the solutions to cubic and quartic equations. One of the equations in his book gave the solution of 5 ± √-15. Commenting on this equation, Cardano wrote, “Dismissing mental tortures, and multiplying 5 + √ – 15 by 5 – √-15, we obtain 25 – (-15). Therefore the product is 40. …. and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless.”

Perhaps the first champion of imaginary numbers was Italian mathematician, Rafael Bombelli (1526-1572). Bombelli understood that i times i should equal -1, and that -i times i should equal one. However, Bombelli could not find a practical use for this property, so he generally was not believed. Bombelli did have what people called a “wild idea” – that imaginary numbers could be used to get real answers.

Imaginary numbers continued to live in disgrace until the work of a series of mathematicians in the 18th and 19th centuries. Leonhard Euler helped clear up some of the problems with using imaginary numbers by developing the notation i to mean √-1. He also introduced the notation a+bi for complex numbers. Carl Friedrich Gauss made imaginary numbers much more concrete and less “imaginary” when he graphed imaginary numbers as points on the complex plane in 1799. However, William Rowan Hamilton in 1833, delivered the coup de grace to imaginary numbers’ bad name when he advanced the idea that complex numbers could be expressed as a pair of real numbers. For example 4+3i could be written simply as (4,3). This made complex numbers much easier to understand and use.

Today, imaginary numbers are an essential part of the everyday calculations that make modern technology work. They are indispensable in the field of electrical engineering, particularly in the analysis of alternating current, like the electrical current that powers household appliances. Also, cell phones and air travel would not be possible without imaginary numbers because they are necessary in the computations involved in signal processing and radar. Imaginary numbers are even used by biologists when studying the firing events of neurons in the brain. Imaginary numbers have come a long way in the five hundred years since they were scoffed at for being absurd and totally useless.